﻿EQUILIBRIUM 
  OF 
  ELASTIC 
  SOLIDS. 
  

  

  95 
  

  

  d 
  8 
  x 
  

   dy 
  

  

  ■m 
  

  

  d8 
  

  

  dz 
  

  

  d 
  8 
  z 
  

   dx 
  

  

  ax 
  

  

  d8 
  z 
  

  

  + 
  oa, 
  = 
  

  

  m 
  

  

  ?3 
  

  

  l-8d,= 
  

  

  -8e 
  a 
  = 
  

  

  + 
  86 
  = 
  —q 
  

  

  dy 
  

  

  d 
  8 
  x 
  $, 
  a 
  1 
  

   dz 
  * 
  m 
  * 
  

  

  (11.) 
  

  

  By 
  substituting 
  in 
  Equations 
  (3.) 
  the 
  values 
  of 
  the 
  forces 
  given 
  in 
  Equa- 
  

   tions 
  (8.) 
  and 
  (9.), 
  they 
  become 
  

  

  d 
  fd8 
  x 
  d 
  8 
  v 
  d 
  8 
  z\\ 
  m 
  / 
  d 
  2 
  <s 
  d 
  2 
  * 
  d 
  2 
  

  

  1 
  N 
  / 
  d 
  fdb 
  x 
  db 
  y 
  d 
  b 
  z\\ 
  m 
  f 
  d 
  2 
  «, 
  <P 
  % 
  d 
  ', 
  2 
  * 
  \ 
  \ 
  

  

  [ 
  f" 
  + 
  6^{jx{-dx- 
  + 
  -df 
  + 
  -dz-)) 
  + 
  T\dl?** 
  + 
  J?** 
  + 
  d?*V 
  + 
  P 
  X 
  = 
  ° 
  

  

  1 
  „ 
  / 
  d 
  fd 
  8 
  x 
  d 
  8 
  y 
  d8 
  z\\ 
  m 
  ( 
  d 
  2 
  » 
  d 
  2 
  % 
  d 
  2 
  «, 
  \ 
  _ 
  r 
  

  

  dy\dx 
  ' 
  dy 
  ' 
  dz) 
  ) 
  ' 
  2\dx 
  2 
  " 
  dy 
  

   d 
  (d 
  8 
  x 
  doy.d8zW.mfd 
  2 
  * 
  d 
  2 
  

  

  (12.) 
  

  

  1 
  . 
  / 
  d 
  fd 
  8 
  x 
  d8 
  y 
  d 
  8 
  z\\ 
  m 
  f 
  d 
  2 
  «, 
  d 
  2 
  » 
  d 
  2 
  * 
  \ 
  „ 
  * 
  

  

  ^ 
  + 
  6 
  m 
  ^{d-z{lT- 
  + 
  ^ 
  + 
  ^) 
  + 
  2{dT 
  28x 
  + 
  dy 
  2 
  ^ 
  + 
  a^ 
  d 
  V 
  + 
  P 
  Z 
  = 
  

  

  These 
  are 
  the 
  general 
  equations 
  of 
  elasticity, 
  and 
  are 
  identical 
  with 
  those 
  

   of 
  M. 
  Cauchy, 
  in 
  his 
  Exercises 
  d' 
  Analyse, 
  vol. 
  hi., 
  p. 
  180, 
  published 
  in 
  1828, 
  when 
  

  

  777/ 
  

  

  k 
  stands 
  for 
  m, 
  and 
  K 
  for 
  fx 
  - 
  -^ 
  , 
  and 
  those 
  of 
  Mr 
  Stokes, 
  given 
  in 
  the 
  Cam- 
  

   bridge 
  Philosophical 
  Transactions, 
  vol. 
  viii., 
  e 
  part 
  3, 
  and 
  numbered 
  (30.) 
  ; 
  in 
  his 
  

   equations 
  A 
  = 
  3/i, 
  B=^. 
  

  

  If 
  the 
  temperature 
  is 
  variable 
  from 
  one 
  part 
  to 
  another 
  of 
  the 
  elastic 
  solid, 
  

   the 
  compressions 
  —r-^> 
  ~~d~^> 
  ~T~~> 
  a 
  ^ 
  an 
  ^ 
  P 
  om 
  * 
  w 
  ^ 
  De 
  diminished 
  by 
  a 
  quan- 
  

   tity 
  proportional 
  to 
  the 
  temperature 
  at 
  that 
  point. 
  This 
  principle 
  is 
  applied 
  in 
  

   Cases 
  X. 
  and 
  XL 
  Equations 
  (10.) 
  then 
  become 
  

  

  d8x 
  ( 
  1 
  l 
  \ 
  . 
  , 
  l 
  \ 
  

  

  d8y 
  f 
  1 
  1 
  \ 
  . 
  . 
  1 
  

  

  -dT 
  = 
  (977-3^) 
  (*+*+*> 
  + 
  «•• 
  + 
  « 
  ^ 
  

  

  d8x 
  

  

  dz 
  

  

  = 
  (977-3^) 
  (ft+A+A)+^* 
  + 
  5 
  A 
  

  

  (13.) 
  

  

  c 
  3 
  # 
  being 
  the 
  linear 
  expansion 
  for 
  the 
  temperature 
  v. 
  

  

  Having 
  found 
  the 
  general 
  equations 
  of 
  the 
  equilibrium 
  of 
  elastic 
  solids, 
  I 
  

   proceed 
  to 
  work 
  some 
  examples 
  of 
  their 
  application, 
  which 
  afford 
  the 
  means 
  of 
  

   determining 
  the 
  coefficients 
  fx, 
  m, 
  and 
  w, 
  and 
  of 
  calculating 
  the 
  stiffness 
  of 
  solid 
  

   figures. 
  I 
  begin 
  with 
  those 
  cases 
  in 
  which 
  the 
  elastic 
  solid 
  is 
  a 
  hollow 
  cylinder 
  

   exposed 
  to 
  given 
  forces 
  on 
  the 
  two 
  concentric 
  cylindric 
  surfaces, 
  and 
  the 
  two 
  

   parallel 
  terminating 
  planes. 
  

  

  vol. 
  xx. 
  part 
  1. 
  2 
  c 
  

  

  